FUNDAMENTAL EQUATIONS OF IDEAL GASES 347 



by simple differentiation. Gibbs has obtained the same result 

 by comparing the terms of the total differential of ip, 



drp = ( -^ ) dt + ( 4- ) dv + ( ^ ) dm, 



\dt/v,m \dv/t.rn \dm/v.t ' 



and 



# = - vdt - pdv + fidm, (11) [88] 



with equation [261]. 



11. f Function for an Ideal Gas. Turning to the zeta 

 function* [91], f = e + pv — trj, we may form the function in 

 terms of pressure, temperature and the mass of a pure perfect 

 gas with the following result : 



f = met + mE + "inat — met log t — mat log — 



- mHt. (12) [265] 



By differentiation the following equations are obtained: 



/9f\ , .at , r , 



- h;7 = V = mclogt -\- ma log— + mH, (13) [266] 



\ot/p,m V 



if) " 



mat , . r 



(14) [267] 



P 

 I ~- I = met + niE + mat = we + mat, (IVb) 



\ OT /pm 



~ ( ^ ) = c + a , 

 m \dtdT/p^m 



/d^\ , , at 



I 7~~ I = n = ct — ct log t — at log — 

 \dm/p, t ^ *^ p 



-\- at - Ht + E. (15) [268] 



* This function is called the "Free Energy" by Lewis and Randall 

 in their treatise Thermodynamics and the Free Energy of Chemical Sub- 

 stances. 



